2. For each of the following, X represents Rx R. Prove that each of the following distance function is a metric on X. Let x = (x1, x₂), y = (91, 92) and z = (21, 22) (i).d(x, y) = √(x₁ - y₁)² + (x2 - y₂)² (Hint: Use the Minkowski's Inequality: Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers. (ii). d(x, y) = x₁y₁| + x2 - y₂| - (iii). d(x, y) = max {|x₁ - y₁|, |x2 - y2|}. 3. Extend each of the above three distance functions to define on X = R" and show that in each case the distance function is a metric.

Q: Let F = (y – x, ry) and let C be the line segment from (2, –3) to (1, –1). %3D Evaluate F. dr.

A: We wil first equation of parametric equation of line segment

Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

A:

Q: a. Find a function (f(x,y,z)) such that (F = Vf(x,y,z))? b. Use part (a) to evaluate (F. dr ) along…

A: We have to evaluate the ∫CF.→dr→.

Q: 2) Let A, B and C be three points on a hyperbolic line such that A* B*C. Let D represent hyperbolic…

A: In hyperbolic geometry, the hyperbolic distance between two points A and B is defined as: D(A, B) =…

Q: The largest set on which the function f(x, y) = 1/(2-x² - y2) is continuous is A. The exterior of…

A:

Q: Let F = Vf and f = 5x²y - 5z. Calculate F. dr for the path r₁ = (t, t,0), 0 ≤ t ≤ 1. (Give your…

A: We will use fundamental theorem of line integrals

Q: 3. Find a point P on the segment AB minimizing the length XP+PY. Hint: denote AP = x and express…

A: Given:

Q: (47) Let C be the curve y = 5x where 1 ≤x≤9. Calculate the length of C.

A:

Q: Consider the curve y = =-2 on the interval [-4-]. Recall that given two points on the curve,(x1F1)…

A: The length of the line segment joining two points is given by, (x2-x1)2+(y2-y1)2=∆x2+∆y2

Q: Let ƒ : R → R be defined by f(x)= x². Calculate each of the following, and circle your final answer.…

A:

Q: Define a metric as d(x,y)=|x1-y1|+|x2-y2|. Find the set of all points x=(x1,x2) in R2 such that…

A: